Given:

 \(\frac{|{\text{x}} |-1}{|{\text{x}} |-2}\) ≥0 x ϵ R. –{–2, 2}

Intervals of |x|:

x ≥ 0, |x| = x and x < 0, |x| = -x

Domain of \(\frac{|{\text{x}} |-1}{|{\text{x}} |-2}\) ≥ 0

\(\frac{|{\text{x}} |-1}{|{\text{x}} |-2}\) is not defined for x = -2 and x = 2

Therefore, Domain: x < -2 or -2 < x < 2 or x > 2 

Combining intervals with domain: 

x < 2, -2<x<0, 0≤x<2, x≤2

For x < -2:

\(\frac{|{\text{x}} |-1}{|{\text{x}} |-2}\) = \(\frac{-{\text{x}} -1}{-{\text{x}} -2}\)

\(\frac{-{\text{x}} -1}{-{\text{x}} -2}\) \(\ge\)0

Signs of – x – 1: 

-x -1 = 0 → x = -1 

(Adding 1 to both the sides and then dividing by -1 on both the sides) 

-x – 1> 0 → x < -1 

(Adding 1 to both the sides and then multiplying by -1 on both the sides) 

-x – 1 < 0 → x > -1 

(Adding 1 to both the sides and then multiplying by -1 on both the sides) 

Signs of – x – 2: 

-x -2 = 0 → x = -2 

(Adding 2 to both the sides and then dividing by -1 on both the sides) 

-x – 2> 0 → x < -2 

(Adding 2 to both the sides and then multiplying by -1 on both the sides) 

-x – 2 < 0 → x > -2 

(Adding 2 to both the sides and then multiplying by -1 on both the sides) 

Intervals satisfying the required condition: ≥ 0 

x < - 2 or x = -1 or x > -1 

Merging overlapping intervals: 

x < -2 or x ≥ -1 

Combining the intervals: 

x < -2 or x ≥ -1 and x < -2 

Merging overlapping intervals: 

x < -2 

Similarly, for -2 < x < 0:

 \(\frac{|{\text{x}} |-1}{|{\text{x}} |-2}\) = \(\frac{-{\text{x}} -1}{-{\text{x}} -2}\)

 \(\frac{-{\text{x}} -1}{-{\text{x}} -2}\) \(\ge\)0

Therefore, 

Intervals satisfying the required condition: ≥ 0 

x < - 2 or x = -1 or x > -1 

Merging overlapping intervals: 

x < -2 or x ≥ -1 

Combining the intervals: 

x < -2 or x ≥ -1 and -2 < x < 0 

Merging overlapping intervals: 

-1 ≤ x < 0 

For 0 ≤ x < 2,

 \(\frac{|{\text{x}} |-1}{|{\text{x}} |-2}\) = \(\frac{-{\text{x}} -1}{-{\text{x}} -2}\)

\(\frac{-{\text{x}} -1}{-{\text{x}} -2}\) \(\ge\)0

Signs of x – 1: 

x – 1 = 0 → x = 1(Adding 1 to both the sides) 

x – 1 > 0 → x > 1(Adding 1 to both the sides) 

x – 1 < 0 → x < 1(Adding 1 to both the sides) 

Signs of x – 2: 

x – 2 = 0 → x = 2(Adding 2 to both the sides) 

x – 2 < 0 → x < 2(Adding 2 to both the sides) 

x – 2 > 0 → x > 2(Adding 2 to both the sides)

At x = 2, ^{\(\cfrac{x-1}{x-2}\)} is not defined

Intervals satisfying the required condition: ≥ 0 

x < 1 or x = 1 or x > 2 

Merging overlapping intervals: 

x ≤ 1 or x > 2 

Combining the intervals: 

x ≤ 1 or x > 2 and 0 ≤ x < 2 

Merging overlapping intervals: 

0 ≤ x ≤ 1 

Similarly, 

for x > 2:

 \(\frac{|{\text{x}} |-1}{|{\text{x}} |-2}\) = \(\frac{-{\text{x}} -1}{-{\text{x}} -2}\)

\(\frac{-{\text{x}} -1}{-{\text{x}} -2}\) \(\ge\)0

Therefore, Intervals satisfying the required condition: ≥ 0 

x < 1 or x = 1 or x > 2 

Merging overlapping intervals: 

x ≤ 1 or x > 2

Combining the intervals: 

x ≤ 1 or x > 2 and x > 2 

Merging overlapping intervals: 

x > 2 

Combining all the intervals: 

x < -2 or -1 ≤ x < 0 or 0 ≤ x ≤ 1 or x >2 

Merging the overlapping intervals: 

x < -2 or -1 ≤ x ≤ 1 or x > 2 

Therefore

x ϵ (-∞, -2) Ս [-1,1] Ս (2, ∞)